The life cycle of a diagonal argument
Diagonal arguments are ubiquitious. We present them in general form, and mention how specific examples arise from this presentation. → read note
Notes
- Research 18
- Mathematics 12
- Diagrammatic Higher Categories 8
- Manifold Diagrams 5
- Learning 3
- Philosophy 3
- Computer Science 3
- Foundations of Mathematics 3
- Geometry 2
Research
Diagrammatic geometry
Diagrammatic geometry makes manifold theory a joyous exercise of diagram-drawing. It has far-reaching consequences for how understanding of the interaction between higher algebra, combinatorics, and differential geometry. → read note
Guide to Kirby diagrams
A brief note about Kirby diagrams, which provide a helpful tool in the representation of 3 and 4-dimensional manifolds → read note
Big vs small things
Big things are different from small things. It’s a fundamental law of mathematical nature. But it’s not so often talked about. Here we at least say once that it’s a thing. (Feel free to extend!) → read note
Diagrammatric observational type theory
Diagrammatic type theory takes a novel approach to working with higher directed inductive types. This does a way with complication of inductive types to unveil a (the) deeper mystery of canonical isotopies. → read note
Diagrammatic mathematics: an overview
Diagrammatic mathematics shines new light on many old problems, ranging from mathematical foundations to specific open problems in geometric topology, and structures from quantum physics. → read note
An invitation to g̶e̶o̶m̶e̶t̶r̶i̶c̶ diagrammatic higher categories (external link) Permalink
Fast and mathematically self-contained introduction to basic ideas in diagrammatic higher category theory, written up as an article for the n-Category Café. → go to link
Mazur manifold as tangle diagram
Mazur manifolds are intimately related to homology spheres, and are part of the puzzling world of 4-manifolds. Here we construct the simplest example of a Mazur manifold as a tangle diagram. → read note
Principles of higher-dimensional logic
We allow ourselves to speculate a bit about how the 1-dimensional logical constructs that one frequently encounters in practical approaches to mathematical foundations may in fact emerge from a set of “more fundamental” principles of compositionality in higher-dimensional logic. → read note
Towards computable manifolds
In this (exceedingly brief) note, we discuss computability questions in the context of manifold theory. The main observation is that several computabilty issues in classical “set-point (differential) topological” foundations may potentially be overcome when working with manifolds from the higher-categorical perspective of manifold diagrams and tangles. → read note
The $D_4$ dualizability law
This note gives a brief visual guide to how one can understand the classical \(D_4\) singularity in terms of a categorical pasting diagram (aka manifold diagram, or more specifically, a tame tangle). → read note
The categorical Pontryagin-Thom construction
The Pontryagin-Thom construction is a deep relation between smooth manifold theory and homotopy theory. We use the language of manifold diagrams and computads to give a purely combinatorial perspective on the construction. On our way, we’ll discuss computadic functors and transformations, and how the combinatorics of manifold diagrams may provide a “combinatorialization” of smooth structures on manifolds altogether. → read note
From trusses to diagrammatic computads to combinatorial manifold diagrams
This note gives a fast introduction to the rich combinatorial theory of trusses. We also discuss how presheafs of truss blocks give rise to diagrammatic computads, and how this can be used to understand manifold diagrams in purely combinatorial terms. → read note
Tame tangles, singularities, and Morse theory
Classical singularity theory and higher category theory have tantalizing connections: via the generalized tangle hypothesis we can understand singularities as expressing categorical laws satisfied by dualizable objects. However, classical singularity theory, based in differential-topological foundations, encounters technical problem in higher dimensions, which stalls this connection. Using our technology of manifold diagrams we outline steps towards remedying the problem, paving the way towards “higher” Morse theory. → read note
Paradigms of higher category theory
We explore two fundamental paradigms underlying models of higher categories. Most models for higher categories are based on the paradigm of contraction; this requires spaces of evaluations of a single pasting diagram to be contractible. The paradigm, however, generally fails to yield easy descriptions of interesting non-contractible behaviour in spaces of composites. This is remedied by working instead with the paradigm of isotopy as we will explain. → read note
Manifold diagrams: a geometric approach
We describe the notion of manifold diagrams. Manifold diagrams generalize string diagrams to higher dimensions. Our focus will lie on giving an intuitively clear geometric description of these diagrams in terms of conical stratifications in a framed background space. We will also provide further details on the dual notion of pasting diagrams of computadic cells. → read note
The class of computadic cells
We describe the problem of defining a “universal” class of cell shapes for higher categories, the central obstruction in the classical approach to the question, and how to overcome that obstruction in the context of framed combinatorial topology. → read note
Framed Combinatorial Topology Permalink
`Framed combinatorial topology’ is the title of a book we just uploaded to the arXiv. On the linked page you’ll find more information! → go to link
Mathematics
An invitation to g̶e̶o̶m̶e̶t̶r̶i̶c̶ diagrammatic higher categories (external link) Permalink
Fast and mathematically self-contained introduction to basic ideas in diagrammatic higher category theory, written up as an article for the n-Category Café. → go to link
Mazur manifold as tangle diagram
Mazur manifolds are intimately related to homology spheres, and are part of the puzzling world of 4-manifolds. Here we construct the simplest example of a Mazur manifold as a tangle diagram. → read note
Towards computable manifolds
In this (exceedingly brief) note, we discuss computability questions in the context of manifold theory. The main observation is that several computabilty issues in classical “set-point (differential) topological” foundations may potentially be overcome when working with manifolds from the higher-categorical perspective of manifold diagrams and tangles. → read note
The $D_4$ dualizability law
This note gives a brief visual guide to how one can understand the classical \(D_4\) singularity in terms of a categorical pasting diagram (aka manifold diagram, or more specifically, a tame tangle). → read note
The categorical Pontryagin-Thom construction
The Pontryagin-Thom construction is a deep relation between smooth manifold theory and homotopy theory. We use the language of manifold diagrams and computads to give a purely combinatorial perspective on the construction. On our way, we’ll discuss computadic functors and transformations, and how the combinatorics of manifold diagrams may provide a “combinatorialization” of smooth structures on manifolds altogether. → read note
From trusses to diagrammatic computads to combinatorial manifold diagrams
This note gives a fast introduction to the rich combinatorial theory of trusses. We also discuss how presheafs of truss blocks give rise to diagrammatic computads, and how this can be used to understand manifold diagrams in purely combinatorial terms. → read note
Tame tangles, singularities, and Morse theory
Classical singularity theory and higher category theory have tantalizing connections: via the generalized tangle hypothesis we can understand singularities as expressing categorical laws satisfied by dualizable objects. However, classical singularity theory, based in differential-topological foundations, encounters technical problem in higher dimensions, which stalls this connection. Using our technology of manifold diagrams we outline steps towards remedying the problem, paving the way towards “higher” Morse theory. → read note
Manifold diagrams: a geometric approach
We describe the notion of manifold diagrams. Manifold diagrams generalize string diagrams to higher dimensions. Our focus will lie on giving an intuitively clear geometric description of these diagrams in terms of conical stratifications in a framed background space. We will also provide further details on the dual notion of pasting diagrams of computadic cells. → read note
The class of computadic cells
We describe the problem of defining a “universal” class of cell shapes for higher categories, the central obstruction in the classical approach to the question, and how to overcome that obstruction in the context of framed combinatorial topology. → read note
Framed Combinatorial Topology Permalink
`Framed combinatorial topology’ is the title of a book we just uploaded to the arXiv. On the linked page you’ll find more information! → go to link
Enriched cohesive infinity toposes
I gave a talk in the Advanced Topology Class at Oxford about “Generalized Differential Cohomology” based on this book. Part of the talk required setting up an “enriched cohesive \(\infty\)-topos”. Here’s an outline of what that object could be and what it does. → read note
Factorization homology
I gave a talk at CUNY about factorization homology, and have written up some notes which I’m posting here. → read note
Diagrammatic Higher Categories
Diagrammatric observational type theory
Diagrammatic type theory takes a novel approach to working with higher directed inductive types. This does a way with complication of inductive types to unveil a (the) deeper mystery of canonical isotopies. → read note
Diagrammatic mathematics: an overview
Diagrammatic mathematics shines new light on many old problems, ranging from mathematical foundations to specific open problems in geometric topology, and structures from quantum physics. → read note
An invitation to g̶e̶o̶m̶e̶t̶r̶i̶c̶ diagrammatic higher categories (external link) Permalink
Fast and mathematically self-contained introduction to basic ideas in diagrammatic higher category theory, written up as an article for the n-Category Café. → go to link
Principles of higher-dimensional logic
We allow ourselves to speculate a bit about how the 1-dimensional logical constructs that one frequently encounters in practical approaches to mathematical foundations may in fact emerge from a set of “more fundamental” principles of compositionality in higher-dimensional logic. → read note
Towards computable manifolds
In this (exceedingly brief) note, we discuss computability questions in the context of manifold theory. The main observation is that several computabilty issues in classical “set-point (differential) topological” foundations may potentially be overcome when working with manifolds from the higher-categorical perspective of manifold diagrams and tangles. → read note
The categorical Pontryagin-Thom construction
The Pontryagin-Thom construction is a deep relation between smooth manifold theory and homotopy theory. We use the language of manifold diagrams and computads to give a purely combinatorial perspective on the construction. On our way, we’ll discuss computadic functors and transformations, and how the combinatorics of manifold diagrams may provide a “combinatorialization” of smooth structures on manifolds altogether. → read note
From trusses to diagrammatic computads to combinatorial manifold diagrams
This note gives a fast introduction to the rich combinatorial theory of trusses. We also discuss how presheafs of truss blocks give rise to diagrammatic computads, and how this can be used to understand manifold diagrams in purely combinatorial terms. → read note
Paradigms of higher category theory
We explore two fundamental paradigms underlying models of higher categories. Most models for higher categories are based on the paradigm of contraction; this requires spaces of evaluations of a single pasting diagram to be contractible. The paradigm, however, generally fails to yield easy descriptions of interesting non-contractible behaviour in spaces of composites. This is remedied by working instead with the paradigm of isotopy as we will explain. → read note
Manifold Diagrams
Mazur manifold as tangle diagram
Mazur manifolds are intimately related to homology spheres, and are part of the puzzling world of 4-manifolds. Here we construct the simplest example of a Mazur manifold as a tangle diagram. → read note
The $D_4$ dualizability law
This note gives a brief visual guide to how one can understand the classical \(D_4\) singularity in terms of a categorical pasting diagram (aka manifold diagram, or more specifically, a tame tangle). → read note
Tame tangles, singularities, and Morse theory
Classical singularity theory and higher category theory have tantalizing connections: via the generalized tangle hypothesis we can understand singularities as expressing categorical laws satisfied by dualizable objects. However, classical singularity theory, based in differential-topological foundations, encounters technical problem in higher dimensions, which stalls this connection. Using our technology of manifold diagrams we outline steps towards remedying the problem, paving the way towards “higher” Morse theory. → read note
Manifold diagrams: a geometric approach
We describe the notion of manifold diagrams. Manifold diagrams generalize string diagrams to higher dimensions. Our focus will lie on giving an intuitively clear geometric description of these diagrams in terms of conical stratifications in a framed background space. We will also provide further details on the dual notion of pasting diagrams of computadic cells. → read note
The class of computadic cells
We describe the problem of defining a “universal” class of cell shapes for higher categories, the central obstruction in the classical approach to the question, and how to overcome that obstruction in the context of framed combinatorial topology. → read note
Learning
Github discussions and writing mathematics (deprecated)
Github markdown does not allow for easily writing mathematical formulaes. We discuss some work-arounds, pairable with the comment plugin giscus. → read note
Enriched cohesive infinity toposes
I gave a talk in the Advanced Topology Class at Oxford about “Generalized Differential Cohomology” based on this book. Part of the talk required setting up an “enriched cohesive \(\infty\)-topos”. Here’s an outline of what that object could be and what it does. → read note
Factorization homology
I gave a talk at CUNY about factorization homology, and have written up some notes which I’m posting here. → read note
Philosophy
Diagrammatic mathematics: an overview
Diagrammatic mathematics shines new light on many old problems, ranging from mathematical foundations to specific open problems in geometric topology, and structures from quantum physics. → read note
Principles of higher-dimensional logic
We allow ourselves to speculate a bit about how the 1-dimensional logical constructs that one frequently encounters in practical approaches to mathematical foundations may in fact emerge from a set of “more fundamental” principles of compositionality in higher-dimensional logic. → read note
Paradigms of higher category theory
We explore two fundamental paradigms underlying models of higher categories. Most models for higher categories are based on the paradigm of contraction; this requires spaces of evaluations of a single pasting diagram to be contractible. The paradigm, however, generally fails to yield easy descriptions of interesting non-contractible behaviour in spaces of composites. This is remedied by working instead with the paradigm of isotopy as we will explain. → read note
Computer Science
The life cycle of a diagonal argument
Diagonal arguments are ubiquitious. We present them in general form, and mention how specific examples arise from this presentation. → read note
Diagrammatric observational type theory
Diagrammatic type theory takes a novel approach to working with higher directed inductive types. This does a way with complication of inductive types to unveil a (the) deeper mystery of canonical isotopies. → read note
Github discussions and writing mathematics (deprecated)
Github markdown does not allow for easily writing mathematical formulaes. We discuss some work-arounds, pairable with the comment plugin giscus. → read note
Foundations of Mathematics
The life cycle of a diagonal argument
Diagonal arguments are ubiquitious. We present them in general form, and mention how specific examples arise from this presentation. → read note
Big vs small things
Big things are different from small things. It’s a fundamental law of mathematical nature. But it’s not so often talked about. Here we at least say once that it’s a thing. (Feel free to extend!) → read note
Diagrammatric observational type theory
Diagrammatic type theory takes a novel approach to working with higher directed inductive types. This does a way with complication of inductive types to unveil a (the) deeper mystery of canonical isotopies. → read note
Geometry
Diagrammatic geometry
Diagrammatic geometry makes manifold theory a joyous exercise of diagram-drawing. It has far-reaching consequences for how understanding of the interaction between higher algebra, combinatorics, and differential geometry. → read note
Guide to Kirby diagrams
A brief note about Kirby diagrams, which provide a helpful tool in the representation of 3 and 4-dimensional manifolds → read note